Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus

In this paper, we study an elliptic system arising from the U(1) \begin{document}$ \times $\end{document} U(1) Abelian Chern-Simons Model[ 25 , 37 ] of the form\begin{document}$ \begin{equation} \left\{\begin{split} \Delta u = &\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $\end{document}which are defined on a parallelogram \begin{document}$ \Omega $\end{document} in \begin{document}$ \mathbb{R}^2 $\end{document} with doubly periodic boundary conditions. Here, \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} are interaction constants, \begin{document}$ \lambda>0 $\end{document} is related to coupling constant, \begin{document}$ m_j>0(j = 1,\cdots,k_1) $\end{document} , \begin{document}$ n_j>0(j = 1,\cdots,k_2) $\end{document} , \begin{document}$ \delta_{p} $\end{document} is the Dirac measure, \begin{document}$ p $\end{document} is called vortex point. Concerning the existence results of this system over \begin{document}$ \Omega $\end{document} , only the cases \begin{document}$ (a,b) = (0,1) $\end{document} [ 28 ] and \begin{document}$ a>b>0 $\end{document} [ 14 ] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters \begin{document}$ (a,b) $\end{document} . We show that the system (1) admits topological solutions provided \begin{document}$ \lambda $\end{document} is large and \begin{document}$ b>a>0 $\end{document} Our arguments are based on a iteration scheme and variational formulation.